Found problems: 106
A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.
Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that
$x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$
Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.
Is there a positive integer $n$ for which the following holds:
for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?
Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number
Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.
Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number
Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.
a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution.
b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as
$$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.
Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that
(a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence);
(b) if the sequence is periodic, then the first number is rational.
(G Shabat)
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$.
(b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$.
Proposed by Karl Czakler
We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that:
a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$
b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers
c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.
(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $
Prove that $\sqrt{1 + xy}$ is a rational number.
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
The function $f : (0,1) \to R$ is defined by
$f(x) = x$ if $x$ is irrational,
$f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$.
Find the maximum value of $f$ on the interval $(7/8,8/9)$.
It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.